A Weyl tensor and coordinate acceleration

[Povel]

TL;DR Summary

Derek Raine's paper "Integral formulation of Mach's Principle" discusses the concept of Weyl curvature and coordinate transformations. Raine argues that when an observer accelerates and sees the universe accelerate in the opposite direction, real Weyl curvature appears that produces inertial forces. This is a Machian explanation for why you feel forces when you accelerate. However, if we start with zero Weyl tensor everywhere, then the Weyl tensor must stay zero when we change coordinates.

I've been reading Derek Raine's paper "Integral formulation of Mach's Principle" from the book "Mach's Principle" by Barbour, and I've hit something that's really bothering me. It seems like there's a problem in how he treats coordinate transformations and the Weyl tensor.

Here are the relevant passages:

In section 3.1 . "Newtonian inertial induction" Raine considers a test body B accelerating through a uniform universe filled with matter at constant density $\rho$. He then switches to B's accelerated reference frame. From B's perspective, the entire universe appears to accelerate in the opposite direction. In this frame, the four-velocity of the cosmic fluid becomes $u^\mu = (1, 0, 0, at)$, which means the energy-momentum tensor $T_{\mu\nu} = \rho u_\mu u_\nu$ picks up off-diagonal components like $T_{0z} = -\rho at$.

Now Raine claims these time-varying components of $T_{\mu\nu}$ generate non-zero Weyl tensor components through Einstein's equations and the Bianchi identities. He specifically calculates $C_{0z0,i} = \frac{1}{2}\kappa\rho a$ and interprets this as showing that "acceleration currents" produce Weyl curvature, which then causes inertial effects - a Machian explanation for why you feel forces when you accelerate.

However, if we start with zero Weyl tensor everywhere (which we should have in a uniform, isotropic universe), then the Weyl tensor has to stay zero when we change coordinates. That's just basic differential geometry - a tensor that's zero in one coordinate system is zero in all of them. So, how can Raine get non-zero Weyl components just from looking at things from an accelerated frame?

The key equation is the contracted Bianchi identity (Raine uses a linearized version of this):$$\nabla^\delta C_{\alpha\beta\gamma\delta} = 8\pi G\left(\nabla_{[\beta}T_{\alpha]\gamma} - \frac{1}{3}g_{\gamma[\alpha}\nabla_{\beta]}T\right)$$

Those are covariant derivatives on the right side. If the matter is just sitting there uniformly and we're only changing our perspective by accelerating, then $\nabla_\sigma T_{\mu\nu}$ stays zero. The covariant derivative is designed specifically to handle coordinate changes properly - the connection terms compensate for any coordinate acceleration.

What makes this even stranger is that this paper was presented at a conference with many experts in general relativity. It seems odd that such a fundamental error - if it is an error - would go unnoticed. This makes me wonder if I'm missing something subtle about the argument.


To think about this more clearly, I came up with the following thought experiment:

Take two concentric spherical shells with uniform density gas between them, with the same density of the shells. Inside the inner shell, place another smaller spherical shell, and within it a charged test particle kept in equilibrium thanks to, for example, some small charge distributed on the second innermost shell.
Now, physically displace the outer shell slightly. The two bigger shells are no longer concentric, which creates a real uniform gravitational field pointing toward the displaced outer shell's centre. The innermost shell starts to fall, and as it accelerates, it creates frame-dragging effects that "pull" the charge along with it. Clearly, when the innermost shell starts falling, the charge inside will also start falling, but I'm imagining that the charge distribution compensates for this, and that by carefully looking at the mutual influence between this distribution and the charge, it is possible to separate the effect of the uniform gravitational acceleration from the frame dragging effect.

This demonstrates that "turning on" a uniform gravitational field has real, observable physical effects - the charge gets dragged along even though it was initially in equilibrium, and presumably this effect takes a certain amount of time to manifest after the initial configuration has been modified. From the particle's perspective at the centre, this should presumably be the same effect it would experience if everything remained static and the particle itself was accelerating through the innermost shell.

This brings me back to Raine's argument. He's essentially claiming that when an observer accelerates and sees the universe accelerate in the opposite direction, real Weyl curvature appears that produces the inertial forces. My thought experiment shows that uniform fields do produce real effects when they're "turned on", and this should be true both in the case when these fields are due to a redistribution of matter (like in the thought experiment) and when they are a result of a change of coordinates, such as that imposed by Raine, since there is locally no difference between the two.
However, the mathematics of general relativity seems to say these are fundamentally different situations - one produces real curvature, albeit temporarily, the other doesn't.

So where's the resolution? Is there an error in how I'm understanding the Bianchi identities in accelerated frames? Is Raine's "post-Newtonian" approximation introducing artefacts that look like real effects?

I'd appreciate any insights on this. The fact that this was presented at a major conference makes me think there might be something I'm not seeing, but I can't figure out what it is.

 

[ergospherical]

I'm not an expert, but that seems sensible. You can't obtain curvature via a coordinate change. The non-zero Weyl component must be a gauge artifact. I imagine he will have spuriously dropped first order terms when trying to linearize the Bianchi identity. If you do it without perturbation theory, or at least make sure to properly expand out and carry all of the first order terms (including the Rindler-frame Christoffels that depend on the acceleration), then you should still get zero Weyl.

 

[Povel]

I'm not an expert, but that seems sensible. You can't obtain curvature via a coordinate change. The non-zero Weyl component must be a gauge artifact. I imagine he will have spuriously dropped first order terms when trying to linearize the Bianchi identity. If you do it without perturbation theory, or at least make sure to properly expand out and carry all of the first order terms (including the Rindler-frame Christoffels that depend on the acceleration), then you should still get zero Weyl.

Thank you.

Out of curiosity, I attempted to perform the calculation in Rindler coordinates.
It boils down to calculating the antisymmetric derivative of the stress-energy tensor. In these coordinates, the Christoffels are not zero, and unsurprisingly, the derivative is zero in the end.

It seems that he obtained a non-null result only because he used Minkowski coordinates to describe the point of view of the accelerated observer (is that even allowed?) and wrote the 4-velocity of the dust like that, including the "at" term.
I also checked that if one considers the 4 velocity that he writes as if the acceleration was the proper one, then his result is recovered.

I think there certainly is a connection between non-null Weyl and frame dragging effect (including linear ones), at least in the case where there is proper acceleration of the source.

From my understanding, though, frame-dragging effects are also expected in cases where the source is undergoing coordinate acceleration, for example, because it is falling toward some far-away mass and the tidal effects are negligible.
This seems to be the case, for example, in the paper "The fourth test of general relativity" by Nordvedt, where on page 10 (640) he describes the following scenario:

The mass elements inside are dragged by the coordinate acceleration, according to the following relation (last equation here, see also eq. 14):

$$\delta \mathbf{a}(\mathbf{r},t) = -\left(2 + 2\gamma + \frac{\alpha_1}{2}\right) \frac{U(\mathbf{r},t)}{c^2} \mathbf{a}$$

Where $\mathbf{a}$ is the coordinate acceleration.

Granted that this calculation is done in the PPN formalism, but still, it would seem to me that this scenario should also involve the Weyl tensor somehow.
However, given how in this case the acceleration is a coordinate one, that can be potentially created by changing coordinates (for example by looking at an inertial shell from the perspective of a properly accelerated observer inside it) I don't know how to reconcile it with the fact the Weyl is a tensor. How does it work in this case?

Given the above, I suppose Raine could be correct in postulating some connection between Weyl and this "coordinate frame dragging", though what he wrote in the book is wrong for the specific case he is examining.

 

[PeterDonis]

frame-dragging effects are also expected in cases where the source is undergoing coordinate acceleration

This can't be right, because frame dragging is an invariant, but coordinate acceleration is not.

This is basically the same argument that @ergospherical gave for why the article cited in the OP can't be right as it stands.

 

[Povel]

This can't be right, because frame dragging is an invariant, but coordinate acceleration is not.

This is basically the same argument that @ergospherical gave for why the article cited in the OP can't be right as it stands.

I understand why Raine's article isn't correct.
However, either I don't understand or I misunderstood some subtleties about Norvedt's article.
Quoting from the article right before and after equation 14:

Suppose one is in the instantaneous rest frame of a celestial body which is being accelerated by an external body. The inductive acceleration term $\frac{\delta h}{\delta t}$ then makes a significant contribution to the inertial reaction of the body which experiences the external acceleration. From (11 c), assuming the body as a whole accelerates at rate $\mathbf{a}$, (13) makes an inertial mass contribution

$$\sum_i M_i \delta a_i = \left(2\gamma + 2 + \frac{\alpha_1}{2}\right) \frac{G}{c^2} \sum_{ij} \frac{M_i M_j}{r_{ij}} \mathbf{a} - \left(1 + \frac{\alpha_2}{2}\right) \frac{G_2}{c^2} \sum_{ij} \frac{M_i M_j}{r_{ij}^3} \mathbf{r}_{ij} \mathbf{r}_{ij} \cdot \mathbf{a}$$


the sum over i,j is over the mass elements of the body. Applied to the earth, (14) makes a contribution to our planet's inertial mass at the level of $3.5 \times 10^{-9}$ of the total mass of the earth (Nordtvedt, 1968, 1982). The lunar laser ranging observations, however, have constrained the gravitational-to-inertial mass ratio of the earth to be 1 to a part in $10^{12}$ (C. O. Alley, personal communication; Williams et al., 1976). The very large contribution to inertial mass due to the gravitational vector potential (14) is needed to reach agreement with observation.

It appears that the accelerations involved here are gravitational in the Newtonian sense (though the same would apply with proper accelerations presumably) and tidal effects due to the attracting body are ignored, so technically these are coordinate accelerations, evaluated from "an instantaneous rest frame".

 

[PeterDonis]

It appears that the accelerations involved here are gravitational in the Newtonian sense

Yes, but there is no such thing in GR.

(though the same would apply with proper accelerations presumably)

No, it wouldn't. Objects undergoing Newtonian "gravitational acceleration" are in free fall. Their proper acceleration is zero. In their own local inertial rest frame, they're just sitting at rest.

 

[PeterDonis]

Norvedt's article

Do you have a link to the article?

 

[PAllen]

A discussion of the absence of the Nordtvedt effect in GR can be found in section 4.3.1 of https://link.springer.com/content/pdf/10.12942/lrr-2014-4.pdf#page47

 

[PAllen]

I think this might be the Nordtvedt article being refernenced. I have no idea if there is a free version of this paper. It is not related to the Nordtvedt effect. Instead, it is an argument that the existence of gravitomagnetsm can be inferred from existing radar ranging experiments.

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.61.2647

 

[Povel]

@PAllen No this is not about Nordtvedt effect. I posted the article link in post 3, here is the link again

https://link.springer.com/article/10.1007/BF00671317

Abstract​

The point of view expressed in the literature that gravitomagnetism has not yet been observed or measured is not entirely correct. Observations of gravitational phenomena are reviewed in which the gravitomagnetic interaction—a post-Newtonian gravitational force between moving matter—has participated and which has been measured to 1 part in 1000. Gravitomagnetism is shown to be ubiquitous in gravitational phenomena and is a necessary ingredient in the equations of motion, without which the most basic gravitational dynamical effects (including Newtonian gravity) could not be consistently calculated by different inertial observers.

I didn't know the article you attached though. thank you

PeterDonis
No, it wouldn't. Objects undergoing Newtonian "gravitational acceleration" are in free fall. Their proper acceleration is zero. In their own local inertial rest frame, they're just sitting at rest.

Sorry, what I meant was that the effect mentioned by Nordtvedt has essentially the same dependencies from the gravitational potential and the acceleration of other cases mentioned in literature where the acceleration was the proper one (caused by a charged shell being attracted or repelled electrically, for example)

 

[Povel]

I found a paper that analyses explicitly the EFE to post-Newtonian order for a frame of reference having an arbitrary time-dependent translational acceleration and an arbitrary time-dependent angular velocity by Nelson

https://link.springer.com/article/10.1007/BF00756150

The resulting equation of motion contains coupled gravitational-inertial terms, see in particular equation 62


$$
\begin{align}
\hat{a}^i &= -\left[1 + \frac{4\phi}{c^2} + \frac{(\mathbf{v} + \boldsymbol{\omega} \times \mathbf{r})^2}{c^2} + \frac{2\mathbf{W} \cdot \mathbf{r}}{c^2}\right] (\text{grad } \phi)^i \nonumber \\
&\quad -[\boldsymbol{\omega} \times (\boldsymbol{\omega} \times \mathbf{r})]^i - \left[\frac{\partial\boldsymbol{\omega}}{\partial t} \times \mathbf{r}\right]^i - 2(\boldsymbol{\omega} \times \mathbf{v})^i \nonumber \\
&\quad -\left(1 + \frac{4\phi}{c^2} + \frac{\mathbf{W} \cdot \mathbf{r}}{c^2}\right)\mathbf{W}^i - \frac{1}{2c^2}[(\mathbf{W} \cdot \nabla)\text{grad } \chi]^i \nonumber \\
&\quad + \frac{1}{c^2}\left\{3\left[\frac{\partial\phi}{\partial t} - (\boldsymbol{\omega} \times \mathbf{r}) \cdot \text{grad } \phi\right] \right. \nonumber \\
&\qquad\qquad + 4(\mathbf{v} + \boldsymbol{\omega} \times \mathbf{r}) \cdot \text{grad } \phi \nonumber \\
&\qquad\qquad + 2(\mathbf{v} + \boldsymbol{\omega} \times \mathbf{r}) \cdot \mathbf{W} \nonumber \\
&\qquad\qquad \left. + \left[\frac{\partial\mathbf{W}}{\partial t} + \boldsymbol{\omega} \times \mathbf{W}\right] \cdot \mathbf{r}\right\} \nonumber \\
&\qquad\qquad \times [\mathbf{v}^i + (\boldsymbol{\omega} \times \mathbf{r})^i] \nonumber \\
&\quad - \frac{1}{c^2}\left\{\frac{\partial\mathcal{U}^i}{\partial t} - [(\boldsymbol{\omega} \times \mathbf{r} \cdot \nabla)\mathcal{U}]^i + (\boldsymbol{\omega} \times \mathcal{U})^i\right\} \nonumber \\
&\quad + \frac{1}{c^2}[(\mathbf{v} + \boldsymbol{\omega} \times \mathbf{r}) \times \overset{\curvearrowright}{\text{curl}} \mathcal{U}]^i \nonumber \\
&\quad - \frac{2}{c^2}(\text{grad } \Psi)^i \tag{62}
\end{align}
$$

Where $\mathbf{v}$ is the velocity measured by the accelerated observer, $\phi$ is the Newtonian gravitational potential, and $\mathbf{W}^i$ is the time-dependent acceleration of the observer's origin of coordinates with respect to the instantaneous rest frame.
The term $\frac{4\phi}{c^2}\mathbf{W}^i$, looks exactly like the leading order term in Nordtvedt equation 14 I wrote in post 5 above, with $\gamma = 1$ and $\alpha_1 = 0$ the values for GR.
It's unclear to me if the second term in Nordtvedt equation 14 can also be recovered from Nelson's paper.

So it seems like an accelerated observer does indeed see a "gravitomagnetic" effect coming from the coordinate accelerated/falling mass.

In the transverse gauge there is a clear relation between the gravitomagnetic "vector potential" $A$, and in particular its time derivative, and the divergence of the traceless tensor potential $d_{ij}$ from the spatial-spatial part of the perturbation, which can also be associated with the electric part of the Weyl tensor.

 

[PAllen]

It seems to me some very different things are being confused. In flat spacetime, there are obviously no coordinates that produce nonzero Weyl curvature. In a GR case like our solar system, all coordinates must show a non vanishing Weyl curvature, since that is invariant. Seeming to me totally separate, is whether gravitomagnetism is detectable in precise solar system measurements. This is what Nordtvedt is arguing. It seems quite plausible to me.

 

[Povel]

@PAllen I think there might be some confusion about what I'm arguing. Let me clarify by summarising all the points I've raised throughout this discussion.

My original concern with Raine's text centers on a basic tensor invariance issue. When I reproduced his calculation using proper Rindler coordinates, the antisymmetric derivative of the stress-energy tensor, and therefore the covariant divergence of the Weyl tensor, correctly yielded zero, as expected. Raine's non-zero result seems to arise from using Minkowski coordinates while treating coordinate acceleration as if it were proper acceleration.

However, this led me to a deeper question about the relationship between coordinate acceleration and gravitomagnetic effects, in particular to the post-Newtonian order, since that seemed to be the scope of Raine's "heuristic" argument linking inertial forces with the Weyl curvature.
I found Nordtvedt's papers, describing scenarios where masses undergoing coordinate acceleration produce frame-dragging effects, with contributions involving their coordinate acceleration and the gravitational potential.
Then I found Nelson's post-Newtonian analysis for arbitrarily accelerated reference frames, where equation 62 cited above contains terms identical to Nordtvedt's leading-order expressions. Nelson's acceleration refers to the coordinate system origin, but an accelerated observer attributes it to the non-accelerated matter with opposite sign, precisely the perspective Raine adopts.

The remarkable pattern I notice is that Raine, despite his flawed approach, recovers effects that depend on coordinate acceleration of matter and the gravitational potential, analogous to both Nordtvedt and Nelson.
Nordtvedt and Nelson certainly work with non-flat metrics, unlike Raine (though he wrote that curvature is "painted" on the flat background, whatever that means), but they all derive similar dependences.

Now, in the transverse gauge, there is a connection between the gravitomagnetic vector potential and the spatial components of the metric perturbation, which link to the electric part of the Weyl tensor via the traceless tensor potential $d_{ij}$. This is discussed in another paper I found

https://www.researchgate.net/public...Scalar_Quantities_in_Gravito-Electromagnetism

The gauge condition linking the two is equation 3.1

$$\frac{2}{c^3} \partial_t A^i + \partial_j \tilde{h}^{ij} = 0$$

with

$$\tilde{h}_{ij} = \frac{2\lambda}{c^4} \eta_{ij} + \frac{2}{c^4} d_{ij}$$

With $\lambda$ a scalar function.

The question then is whether this connection extends to Nelson's terms, which are related to those found by Nordtvedt, and whether this provides a rigorous foundation for the type of effects Raine was attempting to describe.